# rational points on one dimensional sphere

Let

 $\mathbb{S}^{1}=\{(x,y)\in\mathbb{R}^{2}\ |\ x^{2}+y^{2}=1\}$

be a one dimensional sphere.

We will denote by

 $\mathbb{S}^{1}_{\mathbb{Q}}=\{(x,y)\in\mathbb{Q}^{2}\ |\ (x,y)\in\mathbb{S}^{1}\}$

the rational sphere. We shall try to describe $\mathbb{S}^{1}_{\mathbb{Q}}$ in terms of Pythagorean triplets.

Let $(x,y)\in\mathbb{R}^{2}$. Then $(x,y)\in\mathbb{S}^{1}_{\mathbb{Q}}$ if and only if there exists a Pythagorean triplet $a,b,c\in\mathbb{Z}$ (i.e. $|a|,|b|,|c|\in\mathbb{N}$ is a Pythagorean triplet) such that $x,y$ are of the form $\frac{a}{c}$ and $\frac{b}{c}$.

Proof. ,,$\Leftarrow$” If (for example) $x=\frac{a}{c}$ and $y=\frac{b}{c}$ for a Pythagorean triplet $a,b,c\in\mathbb{Z}$, then we have

 $x^{2}+y^{2}=\frac{a^{2}}{c^{2}}+\frac{b^{2}}{c^{2}}=\frac{a^{2}+b^{2}}{c^{2}}=% \frac{c^{2}}{c^{2}}=1$

and thus $(x,y)\in\mathbb{S}^{1}_{\mathbb{Q}}$.

,,$\Rightarrow$” Assume that $(x,y)\in\mathbb{S}^{1}_{\mathbb{Q}}$. Then $x=\frac{p}{q}$ for some $p,q\in\mathbb{Z}$. It follows, that

 $1=x^{2}+y^{2}=\frac{p^{2}}{q^{2}}+y^{2}$

and this is if and only if

 $y=\frac{\sqrt{q^{2}-p^{2}}}{q}$

(up to a sign of course). Therefore $y\in\mathbb{Q}$ if and only if $\sqrt{q^{2}-p^{2}}=n$ is an integer. In this case we have

 $x=\frac{p}{q},\ \ \ \ y=\frac{n}{q}.$

Note, that

 $q^{2}-p^{2}=n^{2}$

and thus

 $q^{2}=n^{2}+p^{2},$

so $n,p,q\in\mathbb{Z}$ is a Pythagorean triplet, which completes the proof. $\square$

Corollary. The rational sphere $\mathbb{S}^{1}_{\mathbb{Q}}$ is an infinite set.

Proof. Let $m,n\in\mathbb{N}$ be natural numbers such that $n$ is fixed and even. Let $m$ run through primes. Then (due to the theorem in parent entry)

 $2mn,\ m^{2}-n^{2},\ m^{2}+n^{2}\in\mathbb{N}$

is a Pythagorean triplet. Let

 $x_{m}=\frac{2mn}{m^{2}+n^{2}}=\frac{2n}{m+\frac{n^{2}}{m}}.$

Ir follows from the theorem, that there exists $y_{m}\in\mathbb{Q}$ such that $(x_{m},y_{m})\in\mathbb{S}^{1}_{\mathbb{Q}}$. It is easy to see, that $x_{m}=x_{m^{\prime}}$ if and only if $m=m^{\prime}$ and thus we generated infinitely many rational points on sphere. This completes the proof. $\square$

Title rational points on one dimensional sphere RationalPointsOnOneDimensionalSphere 2013-03-22 19:07:49 2013-03-22 19:07:49 joking (16130) joking (16130) 7 joking (16130) Definition msc 11-00 RationalSineAndCosine